6.2. estimating phase distribution of contaminants in model worlds

6.2. Estimating phase distribution of contaminants in model worlds

EPEnvironmental Processes

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Aims and Outcomes

Aims: i. to provide overview of main transport mechanisms in all

environmental compartments ii. to give information about methods of estimation of distribution of

pollutants in the environment Outcomes: iii. students will be able to estimate main transport mechanisms of real

pollutants on the base of their physical-chemical properties iv. students will be able to estimate the distribution of pollutants in the

environment on the base of environmental models

Environmental processes/6-2/Estimating phase distribution of contaminants in model worlds

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Lecture Content

• Description of basic transport mechanisms of pollutants in environmental compartments (diffusion, dispersion, advection)

• Definition of fugacity• Multi-media fugacity models (level I, II, III)

Content of the practical work:1. Transport in porous media.2. Transport through boundaries (bottleneck/wall and diffusive

boundaries)


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Compartment system

• The whole environment is highly structured• Simplification for modeling: compartment system

– Compartment• Homogeneously mixed• Has defined geometry, volume, density, mass, …

• Closed and open systems

Compartment 1

Compartment 2 Compartment 3

Closedsystem

Compartment 1

Compartment 2

Compartment 3

Opensystem


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Transport Mechanisms in the Environment• Diffusion

– movement of molecules or particles along a concentration gradient, or from regions of higher to regions of lower concentration.

– does not involve chemical energy (i.e. spontaneous movement)

Fick’s First Law of Diffusion:

xCDAAJN diffdiff

Ndiff … net substance flux [kg.s-1]Jdiff … net substance flux through the unit

area [kg s-1 m-2]A … cross-sectional area (perpendicular to

diffusion) [m2]D … diffusion coefficient [m2 s-1]�C/x … concentration gradient [kg m-3 m-1]


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Transport Mechanisms in the Environment• Diffusion (contd.)

– Fick’s First Law of Diffusion is valid when:• The medium is isotropic (the medium and diffusion

coefficient is identical in all directions)• the flux by diffusion is perpendicular to the cross section area• the concentration gradient is constant

– Usual values of D:• Gases: D 10-5 - 10-4 m2 s-1

• Liquids: D 10-9 m2 s-1

• Solids: D 10-14 m2 s-1

Barrow, G.M. (1977): Physikalische Chemie Band III. Bohmann, Wien, Austria, 3rd ed.


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Transport Mechanisms in the Environment• Diffusion coefficient (or diffusivity)

– Proportional to the temperature– Inversely proportional to the molecule volume (which is related

to the molar mass)– Relation between diffusion coefficients of two substances:

Tinsley, I. (1979): Chemical Concepts in Pollutant Behaviour. John Wiley & Sons, New York.

i

j

j

i

M

MDD

Di, Dj … diffusion coefficients of compounds i and j [m2 s-1]Mi, Mj … molar masses of compounds i and j [g mol-1]


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Transport Mechanisms in the Environment• Diffusion conductance (g), diffusion resistance (r)

xD

rg

1 g … diffusion conductance [m s-1]

r … diffusion resistance [s m-1]D … diffusion coefficient [m2 s-1]x … diffusion length [m]

More than 1 resistance in system calculation of total resistance using Kirchhoff laws

Resistances in series: 𝒓 𝒕𝒐𝒕𝒂𝒍=𝒓𝟏+𝒓𝟐+…+𝒓𝒏

Resistances in parallel: 𝒈𝒕𝒐𝒕𝒂𝒍=𝒈𝟏+𝒈𝟐+…+𝒈𝒏


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Transport Mechanisms in the Environment• Fick Second Law of Diffusion:

𝜕𝐶𝜕𝑡 =𝐷 𝜕2𝐶

𝜕𝑥2

For three dimensions:

𝜕𝐶𝑑𝑡 =𝐷𝑥

𝜕2𝐶𝜕𝑥2 +𝐷 𝑦

𝜕2𝐶𝜕𝑦2 +𝐷𝑧

𝜕2𝐶𝜕𝑧2

Dx, Dy, Dz … diffusion coefficients in x, y and z direction


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Transport Mechanisms in the Environment• Dispersion:

– Random movement of surrounding medium in one direction (or in all directions) causing the transport of compound

– Mathematical description similar to diffusion

xCDAAJN dispdispdisp

Ndisp … net substance flux [kg.s-1]Jdisp … net substance flux through the unit

area [kg s-1 m-2]A … cross-sectional area (perpendicular to

dispersion direction) [m2]Ddisp … dispersion coefficient [m2 s-1]�C/x … concentration gradient [kg m-3 m-1]

𝜕𝐶𝜕𝑡 =𝐷𝑑𝑖𝑠𝑝

𝜕2𝐶𝜕𝑥2


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Transport Mechanisms in the Environment• Advection (convection):

– the directed movement of chemical by virtue of its presence in a medium that happens to be flowing

CuAAJN advadv Nadv … net substance flux [kg.s-1]Jadv … net substance flux through the unit

area [kg.s-1.m-2]A … cross-sectional area (perpendicular to

u) [m2]u� … flow velocity of medium [m.s-1]

𝜕𝐶𝜕𝑡 =

𝐴𝑉 𝑢 ∙𝐶

𝜕𝐶𝜕𝑡 =−𝑢 ∙ 𝜕𝐶𝜕𝑥


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Chemical reaction

– Process which changes compound’s chemical nature (i.e. CAS number of the compound(s) are different)

Zero order reaction • reaction rate is independent on the concentration of parent compounds

𝑑𝐶𝑑𝑡 =−𝑘0

𝐶𝑡=𝐶0−𝑘0 ∙ 𝑡

k0 … zero order reaction rate constant [mol.s-1]

C0 … initial concentration of compound [mol.L-1]

Ct … concentration of compound at time t [mol.L-1]


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Chemical reaction

First order reaction:• Reaction rate depends linearly on the concentration of one parent compound

𝑑𝐶𝑑𝑡 =−𝑘1 ∙𝐶

𝐶𝑡=𝐶0𝑒−𝑘1 ∙𝑡

k1 … first order reaction rate constant [s-1]C0 … initial concentration of compound

[mol.L-1]Ct … concentration of compound at time t

[mol.L-1]


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Chemical reaction

Second order reaction:• Reaction rate depends on the product of concentrations of two parent compounds

𝑑𝐶𝐴

𝑑𝑡 =−𝑘2 ∙𝐶𝐴∙𝐶𝐵k2 … second order reaction rate constant of

compound A [mol˗1.s-1]CA, CB … initial concentration of compounds A

and B [mol.L-1]

Pseudo-first order reaction:Reaction of the second order could be expressed as pseudo-first order by multiplying the second order rate constant of compound A with the concentration of compound B:

𝑘1 ,𝐴=𝑘2 ∙𝐶𝐵k2 … pseudo-first order reaction rate constant

of compound A [s-1]


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Chemical reaction

Michaelis-Menten kinetics:• Takes place at enzymatic reactions • Reaction rate v [mol.L-1] depends on

• enzyme concentration• substrate concentration C• affinity of enzyme to substrate Km

(Michaelis-Menten constant)• maximal velocity vmax

𝒗=𝒗𝒎𝒂𝒙 ∙𝑪𝑲𝒎+𝑪

When C << Km approx. first order reaction (transformation velocity equal to C)When C >> Km approx. zero order reaction (transformation velocity independent on C)


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Fugacity

• Fugacity – symbol f - proposed by G.N. Lewis in 1901– From Latin word “fugere”, describing escaping tendency of

chemical– In ideal gases identical to partial pressure– It is logarithmically related to chemical potential– It is (nearly) linearly related to concentration

• Fugacity ratio F: – Ratio of the solid vapor pressure to supercooled liquid vapor

pressure– Estimation: 𝐥𝐨𝐠 𝑭=−𝟎 .𝟎𝟏 (𝑻𝑴−𝟐𝟗𝟖 ) TM … melting point [K]


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Fugacity

• Fugacity capacity Z

Gas phase: 𝒁 𝑨=𝑪 𝑨

𝒇ZA … fugacity capacity of air [mol.m-3.Pa-1]CA … air concentration [mol.l-1]f … fugacity [Pa]

Water phase: 𝒁𝑾=𝟏𝑯

ZW … fugacity capacity of water [mol.m˗3.Pa-1]

H … Henry’s law constant [Pa.m3.mol-1]


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Multimedia Environmental Models

Reason for the using of environmental models:• Possibility of describing the potential distribution and environmental

fate of new chemicals by using only the base set of physico-chemical substance properties

• Their use recommended e.g. by EU Technical Guidance Documents– multi-media model consisting of four compartments

recommended for estimating regional exposure levels in air, water, soil and sediment.• Technical Guidance Documents in Support of The Commission Directive

93/67/EEC on Risk Assessment For New Notified Substances and the Commission Regulation (EC) 1488/94 on Risk Assessment For Existing Substances


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Classification of environmental models:• Level 1: Equilibrium, closed system, no reactions• Level 2: Equilibrium, open system, steady state, reactions• Level 3: Non-equilibrium, open system, steady-state• Level 4: Non-equilibrium, open system, non-steady state.


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Multimedia Environmental ModelsEnvironmental Models Level 1: Closed system, equilibrium, no reactions

Com

part

men

t 1

Com

part

men

t 2

Com

part

men

t 3

Total mass in system: m [kg]Volumes of compartments Vn [m3]Unknown concentrations Cn

𝒎=𝑪𝟏 ∙𝑽𝟏+𝑪𝟐 ∙𝑽𝟐+…+𝑪𝒏 ∙𝑽 𝒏

In equilibrium:

𝐶𝑖

𝐶1=𝐾 𝑖 ,1 i = 1, …, n

𝑪𝟏=𝒎

𝑽 𝟏+𝑲𝟐 ,𝟏 ∙𝑽𝟐+…+𝑲𝒏 ,𝟏 ∙𝑽 𝒏

𝑪𝒊=𝑲 𝒊 ,𝟏 ∙𝑪𝟏

𝒎𝒊=𝑽 𝒊 ∙𝑪 𝒊


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Environmental Models Level 2: Equilibrium with source and sink, steady-state, no reactions

Com

part

men

t 1

Com

part

men

t 2

Com

part

men

t 3

INPUT

OUTPUT

Steady-state:

𝒅𝒎𝒅𝒕 =𝟎

Input = Output

Advection into the system [mol.s-1] : I = Q . C Q … flow [m3.s-1]C … concentration [mol.m-3]

Advection out of the system:

𝑂=∑𝑖

(𝑉 𝑖 ∙𝐶𝑖∙ 𝑖 ) I … elimination rate (first order rate), flux per volume

𝑖=𝑄𝑉


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Environmental Models Level 2: Equilibrium with source and sink, unsteady state, no reactions

𝒅𝒎𝒅𝒕 =𝒊𝒏𝒑𝒖𝒕−𝒐𝒖𝒕𝒑𝒖𝒕

𝑑𝑚𝑑𝑡 =∑

𝑖𝐼 𝑖−∑

𝑖(𝑉 𝑖 ∙𝐶𝑖 ∙𝑖 )

In equilibrium:𝐶𝑖

𝐶1=𝐾 𝑖 ,1 i = 1, …, n

𝑑𝑚𝑑𝑡 =𝑉 1

𝑑𝐶1

𝑑𝑡 +𝑉 2

𝑑𝐶2

𝑑𝑡 +…+𝑉 𝑛𝑑𝐶𝑛

𝑑𝑡

𝑑𝑚𝑑𝑡 =𝑉 1

𝑑𝐶1

𝑑𝑡 +𝐾 2,1 ∙𝑉 2

𝑑𝐶1

𝑑𝑡 +…+𝐾 𝑛 , 1 ∙𝑉𝑛𝑑𝐶1

𝑑𝑡


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Environmental Models Level 2: Equilibrium with source and sink, non-steady state, no reactions (cont.)

𝑑𝐶1

𝑑𝑡 =∑𝑖𝐼 𝑖−𝐶1∑

𝑖(𝑉 𝑖 ∙𝐾 𝑖 , 1∙𝑖 )

𝑉 1+𝐾 2,1∙𝑉 2+…+𝐾𝑛 ,1 ∙𝑉 𝑛

or𝑑𝐶1

𝑑𝑡 =−𝑎 ∙𝐶1+𝑏

𝑎=∑𝑖

(𝑉 𝑖 ∙𝐾 𝑖 ,1 ∙𝑖)

𝑉 1+𝐾 2,1∙𝑉 2+…+𝐾𝑛 , 1 ∙𝑉 𝑛𝑏=

∑𝑖𝐼𝑖

𝑉 1+𝐾 2,1 ∙𝑉 2+…+𝐾 𝑛 ,1 ∙𝑉 𝑛

Solution for C1(t): 𝑪𝟏 (𝒕 )=𝒆−𝒂𝒕+𝒃𝒂 (𝟏−𝒆−𝒂𝒕 )


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Environmental Models Level 3: • No equilibrium, sources and sinks, steady state, degradation. • For every single compartment input and/or output may occur. • The exchange between compartments is controlled by transfer

resistance.

Com

part

men

t 1

Com

part

-m

ent 2

Com

part

-m

ent 3

INPUT 1

OUTPUT 2

INPUT 2

OUTPUT 1


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Environmental Models Level 3 (contd.):

𝒅𝒎𝒊

𝒅𝒕 =𝑽𝒊

𝒅𝑪 𝒊

𝒅𝒕 =𝑰 𝒊+𝑵 𝒊+∑𝒋

(𝑵 𝒊𝒋 )−𝑪𝒊 ∙𝑽 𝒊 ∙𝒊=𝟎

Change of substance mass in compartment (i) = Input Ii + advective transport Ni + diffusive transport Nij – output = 0 (steady state)


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Environmental Models Level 4: • No equilibrium, sources and sinks, unsteady state, degradation. • For every single compartment input and/or output may occur. • The exchange between compartments is controlled by transfer

resistance.

𝒅𝒎𝒊

𝒅𝒕 =𝑽𝒊

𝒅𝑪 𝒊

𝒅𝒕 =𝑰 𝒊+𝑵 𝒊+∑𝒋

(𝑵 𝒊𝒋 )−𝑪𝒊 ∙𝑽 𝒊 ∙𝒊≠𝟎

Change of substance mass in compartment (i) = Input Ii + advective transport Ni + diffusive transport Nij – output 0 (unsteady state)


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Further reading

• D. Mackay: Multimedia environmental models: the fugacity approach. Lewis Publishers, 2001, ISBN 978-1-56-670542-4

• S. Trapp, M. Matthies: Chemodynamics and environmental modeling: an introduction. Springer, 1998, ISBN 978-3-54-063096-8

• L. J. Thibodeaux: Environmental Chemodynamics: Movement of Chemicals in Air, Water, and Soil. J. Wiley & Sons, 1996, ISBN 978-0-47-161295-7

• M.M. Clark: Transport Modeling for Environmental Engineers and Scientists. J. Wiley & Sons, 2009, ISBN 978-0-470-26072-2

• C. Smaranda and M. Gavrilescu: Migration and fate of persistent organic pollutants in the atmosphere - a modelling approach. Environmental Engineering and Management Journal, 7/6 (2008), 743-761


6.2. estimating phase distribution of contaminants in model worlds

Documents

distribution of pollutants

law of diffusion

j m2 s

cx concentration gradient

regions of lower concentration

j g mol

constantusual values

jdiff net substance